__Pressure Tendency Equation__

__ __

The Equation of Continuity is

(1)

where is the three-dimensional wind vector.

The hydrostatic equation is

(2)

Integrate (2) from a level z1 where the
pressure is p1=p to the top of the atmosphere at level z2 where p2=0.

(3)

Equation (3) states that the
pressure at any level is directly proportional to the density of the
atmospheric layer of thickness **dz
** or, in other words, the weight of the slab of atmosphere of
thickness **dz**. If level 1 is sealevel and level 2 is
the top of the atmosphere, then equation (3) simply states that sealevel
pressure is really a function of the
total weight of the atmospheric column.

The local pressure tendency
can be determined from (3) for a given air column with dz thickness (dz is
treated as constant) and is given as:

(4)

Substitute (1) into (4) to
obtain the relationship between pressure tendency at a given level to the mass
divergence with respect to and the mass advection in/out of the air column.

(5)

Local Mass Mass

Pres due
to In/Out
and Divergence Or

Tend Advection Convergence

At the synoptic scale, the
advection of mass is small and may be dropped from this equation on an order of
magnitude basis. (*Note that
this term is NOT small near fronts, outflow boundaries, sea-breeze fronts and
other mesoscale features and MUST be retained in understanding pressure
development with respect to those features.)*

* *

Thus, for synoptic scale flow, Equation (5)
reduces to

* ** *(6)

Substitution of * *. into (6) gives

(7)

which states that the
pressure tendency at the base of an air column is a function of horizontal mass
divergence into/out of the air column AND the vertical mass transport through
the top and bottom of the air column.

The term * *represents difference in vertical wind through the air
column defined by the depth dz. As
such, the finite differencing yields

* *(8)

If the top of the air column
is taken as always at the top of the atmosphere (or troposphere), then w2 = 0 which
further simplifies the expression when substituted into (8) and if (8) is
substituted into (7).

* ** *(9)

Equation (9) is the general
pressure tendency equation. This
states that the change in pressure
observed at the bottom of a slab of air of thickness dz is due to the net
horizontal divergence in/out of the slab modified by the amount of mass being
brought in through the bottom of the slab. In this case, the top of the slab is also the top of the
atmosphere. See Fig. 1.

*Figure
1*

* *

To make this equation
"relevant" to the surface weather map, remember that at the ground, w1=0. Hence, the SURFACE PRESSURE TENDENCY
EQUATION is:

* *(10)

where * *is the *NET HORIZONTAL DIVERGENCE[1]* in the air column from the ground to the top of the
atmosphere and can be approximated by the product of the NET DIVERGENCE
(obtained by summing all the horizontal divergences through the layer) and the
thickness of the layer.
Remembering that 90% of the mass of the atmosphere lies beneath the
tropopause and that we are treating the density as a mean density for the air
column (a constant), then it can be seen that, at a synoptic-scale, surface
pressure tendencies are directly related to horizontal divergence patterns in
the troposphere.

Equation (10) can be used
computationally to obtain qualitative estimates of surface pressure
development. Substitution of
equation (2) into the right side of (10) eliminates the density and gravity and
allows one to compute the surface pressure tendency on the basis of the net
divergence through the layer of **dp**
thickness.

__DineÕs Compensation__

__ __

The Equation of Continuity is

(1)

where is the three-dimensional wind vector.

At the synoptic-scale, gross
scaling allows us to drop the local tendency, and the advection of
density. Equation (1) reduces to

(2)

which can be rewritten

(3)

Equation (3) is the so-called ÒDineÕs CompensationÓ principle, where the left hand side can be thought of as the ÒnetÓ horizontal divergence in the layer of Æz thickness (where the Æw is estimated from top to bottom of layer), but is really the Equation of Continuity simplified for gross synoptic-scale conditions.

Recall that the Pressure Tendency Equation is

* ** * (4)

If one assumes that the local
pressure tendency is negligible (drops on an order of magnitude basis—not
true for small time frames), then equation (4) is the same as equation
(3). In Fig. 1, the left hand side
of (3) would be the net divergence at A.

[1] Please note that the net divergence is not the mean divergence. The net divergence represents the arithmetic sum of the divergence from each layer from the bottom to the top of the slab considered.